Extracting work from a single heat bath: A case study of a Brownian particle under an external magnetic field in the presence of information

Work can be extracted from a single bath beyond the limit set by the second law of thermodynamics by performing measurement on the system and utilizing the acquired information. This imposes an upper bound on extracted work and maintains a generalized (i.e., with information) second law. As an example, we studied a Brownian particle confined in a two-dimensional harmonic trap in the presence of a magnetic field, whose position coordinates are measured with finite precision. Two separate cases are investigated in this study: (A) moving the center of the potential and (B) varying the stiffness of the potential. Optimal protocols that extremize the work in a finite-time process are explicitly calculated for these two cases. For case A, we show that even though the optimal protocols depend on magnetic field, surprisingly, extracted work is independent of the field. For case B, both the optimal protocol and the extracted work depend on the magnetic field. However, the presence of a magnetic field always reduces the extraction of work for the latter case.

Figure 1

Optimal protocols corresponding to moving trap are plotted for cyclic process; i.e., $f_1=f_2=0$ in the presence of measurement for total time $t=1$. The error width is taken to be $\sigma =0.4$. In (a) and (b), the optimal protocols ${\alpha }_x^{*}$ and ${\alpha }_y^{*}$ are plotted for fixed magnetic field $C=5$ but for three different measurement outcomes. In (c) and (d) the same are plotted with a fixed measurement outcome, $x_m=y_m=1.5$, but with four different magnetic fields.

Figure 2

Average extremal work for moving trap (a) with measurement (error width $\sigma =0.4$), (b) without measurement. In (a) we plot the extremal work for two different final conditions, its asymptotic value for $t\to \infty$ limit, and the acquired information $I$ due to measurement.

Figure 3

Optimal protocol and work for stiffness varying trap are plotted in the presence of measurement for cyclic process in total time $t=1$. The process being cyclic, the initial and final stiffness are the same and chosen to be unity. In (a) and (c), optimal protocol and work are plotted for fixed magnetic field $C=5$, whereas in (b) and (d) are plotted for fixed error width $\sigma =0.4$. In (c) and (d), log time time scale is used to depict the fact that $W^{*}$ saturates at large times.

Figure 4

Plot of extracted work in the units of $k_{B}T$ at $t\to \infty$ limit for case A and case B.